ENTRY ABSTRACT ALGEBRA
[ENTRY ABSTRACT ALGEBRA] Authors: started Oliver Knill: September 2003 Literature: Lecture notes
additive
A function f : G H from a semigroup G to a semigroup H is [additive] if f(a + b) = f(a) + f(b). A group-valued function→ on sets is additive if f(Y Z) = f(Y ) + f(Z) if Y and Z are disjoint. ∪
algebra
An [algebra] over a field K is a ring with 1 which is also a vector space over K and whose multiplication is bilinear with respect to K. Examples:
the complex numbers C is an algebra over the field of real numbers K = R. • The quaternion algebra H is an algebra over the field of complex numbers. • The matrix algebra M(n, R) is an algebra over the field R. •
An algebraic number field
[An algebraic number field] is a subfield of the complex numbers that arises as a finite degree algebraic extension field over the field of rationals.
alternating group
The [alternating group] G is the subgroup of the symmetric group of n objects given by the elements which can be written as a product of an even number of transpositions.
Artinian module
An [Artinian module] is a module which satisfies the descending chain condition. Every Artinian module is a Noetherian module but the integers for example are a Noetherian module which is not an Artinian module.
Artinian ring
An [Artinian ring] is a ring which when considered as a R-module is an Artinian module.
Artinian ring
Two elements of an integral domain that are unit-multipliers of each other are called [associate numbers]. Cayley’s theorem
[Cayley’s theorem] assures that every finite group is isomorphic to a permutation group.
The [center] of a group (G, ) is the set of all elements g which satisfy gh = hg for all h in G. The center is a subgroup of G. ∗
commutator
1 1 The [commutator] of two elements g, h in a group (G, ) is defined as [g, h] = g h g− h− . ∗ ∗ ∗ ∗
commutator subgroup
The [commutator subgroup] of a group (G, ) is the set of all commutators [g, h] in G. It is a subgroup of G. ∗
factor group
A [factor group] G/N is defined when N is a normal subgroup of the group G. It is the group, where the elements are equivalent classes gN and operation (gN)(hN) = (gh)N which is defined because N was assumed to be normal. For example, if G is the group of additive integers and N = kN with an integer k, then G/N = Zk is finite group of integers modulo k.
finite group
A group is called a [finite group] if G is a set with finitely many elements. For example, the set of all permutations of a finite set form a finite group. The set of all operations on the Rubik cube form a finite group. group
A [group] (X, +, 0) is a set X with a binary operation + and a zero element 0 (also called neutral element or identity) with the following properties
(a + b) + c = a + (b + c) associativity a + 0 = a zero element a ba + b = 0 inverse ∀ ∃ Examples:
the real numbers form a group under addition 5 + 2.34 = 7.34, 3 3 = 0. • −
the set GL(n, R) of real matrices with nonzero determinant form a group under matrix multiplication • the nonzero integers form a group under multiplication 4 7 = 28. • ∗
all the invertible linear transformations of the plane plane form a group under composition. The ”zero • element” is the identity transformation T (x) = x.
all the continuous functions on the unit interval form a group with addition (f + g)(x) = f(x) + g(x). • all the permutations on a finite set form a group under composition. • the set of subsets Y of a set X with the operation A∆B = (A B) (A B) form a group. The inverse • of A is A itself because A∆A = , the zero element is . ∪ \ ∩ ∅ ∅
normal subgroup a [normal subgroup] of a group (G, ) is a subgroup (H, ) of (G, ) which has the property that for all g in H 1 ∗ ∗ ∗ and all g in G one has g− hg is in H. For an abelian group all subgroups are normal. The subgroup Sl(n, R) of Gl(n, R) is a normal subgroup.
ring
A [ring] (X, +, , 0) is a set X with a binary operation + and a binary operation such that (X, +, 0) is a commutative group∗ and (X, ) is a semigroup and such that the distributivity laws ∗a (b + c) = a b + a c, (a + b) c a c + b c hold.∗ Examples: ∗ ∗ ∗ ∗ − ∗ ∗ the integers Z form a ring with addition and multiplication • the set of rational numbers Q, the set of real numbers R or the complext numbers C form a ring with • addition and multiplication. the set of 3x3 matrices with real entries form a ring with addition and matrix multiplication. • the set P of polynomials with real coefficients form a ring with addition and multiplication. • the set of subsets Y of a set X with addition ∆ and multiplication forms a ring. • ∩ the set of continuous functions on an interval [0, 1] with addition (f +g)(x) = f(x)+g(x) and multiplication • f g(x) = f(x)g(x). ∗ commutative group
A [commutative group] is a group (X, +, 0) which is commutative: a + b = b + a.
the set of real numbers R forms a commutative group under addition. • the set of permutations S of a set X form a noncommutative group under composition. •
A [commutative ring] is a ring (X, +, , 0) for which the multiplicative semigroup (X, ) is commutative: a b = b a. Examples: ∗ ∗ ∗ ∗ the integers form a commutative ring. • the set of 2 2 matrices form a noncommutative ring • × the set of polyomials with real coefficients (x2 + πx + 2) (x + 5x) = 6x3 + 6πx2 + 12x. • ∗
function field
A [function field] is a finite extension of the field C(z) of rational functions in the variable z.
An [homomorphism] φ between two groups G, H is a map f : G H which has the property φ(g h) = φ(g) φ(h) and φ(0) = 0 for all elements g, h G. Examples: → ∗ ∗ ∈ if G is the multiplicative group (R+, ) of positive real numbers and H is the additive group (R, +) of all • positive real numbers then φ(x) = log∗(x) is a homomorphism:
if G is the group of matrices with nonzero determinant and H is the group of nonzero real numbers and • φ(A) = det(A), we have φ(x y) = φ(x)φ(y). ∗
An [isomorphism] φ between two groups G, H is a homomorphism between groups which is also invertible. number field
A [number field] is a finite extension of Q, the field of rational numbers. It is a field extension of Q which is also a vector space of finite dimension over Q. Since the elements of a number field are algebraic numbers, roots of n a fixed polyonomial a0 + a1z + ... + z with integer coefficitients, one calls them also algebraic number fields. The study of algebraic number fields is part of algebraic number theory. Examples:
quadratic fields: Q(√d), where d is a rational number. It is in general a field extension of degree 2 over • the field of rational number.
cyclotimic fields: Q(ξ), where ξ is a n’th root of 1. It is a field extension of degree φ(n), where φ(n) is the • Euler function.
The [octonions] can be written as linear combinations of elements e0, e1, e2, ..., e7. The multiplication is deter- mined by the multiplication table
* 1 e1 e2 e3 e4 e5 e6 e7
1 1 e1 e2 e3 e4 e5 e6 e7 e e 1 e e -e e -e -e 1 1 − 4 7 2 6 5 3 e e -e 1 e e -e e -e 2 2 4 − 5 1 3 7 6 e e -e -e 1 e e -e e 3 3 7 5 − 6 2 4 1 e e e -e -e 1 e e -e 4 4 2 1 6 − 7 3 5 e e -e e -e -e 1 e e 5 5 6 3 2 7 − 1 4 e e e -e e -e -e 1 e 6 6 5 7 4 3 1 − 2 e e e e -e e -e -e 1 7 7 3 6 1 5 4 2 − Octonions are also called Cayley numbers. The multiplication of octonions is not associative. Octonions have been discovered by John T. Graves in 1843 and independently by Arthur Cayley.
The [order] of a finite group is the set of elements in the group.
p-group
A [p-group] is a finite group with order pn, where p is a prime integer and n > 0. quaterions
The [quaterions] can be written as linear combinations of elements 1, i, j, k. The multiplication is determined by the multiplication table * 1 i j k 1 1 i j k i i 1 k j − − j j k 1 i − − k k j i 1 − − Quaternions are useful to compute rotations in three dimensions.
semigroup
A [semigroup] (X, +) is a set X with a binary operation + which satisfies the associativity law (a + b) + c = a + (b + c). Examples:
a group is a semigroup. • the set of finite words in an alphabet with composition form a semigroup word1 + word2 = word1word2 • the natural numbers form a semigroup under addition. •
commutative semigroup
A [commutative semigroup] is a semigroup (X, +) which is commutative. a + b = b + a.
the natural numbers form a commutative semigroup under addition. • composition of words over a finite alphabet form a noncommutative semigroup •
The [kernel] of a homomorphism between two groups G, H is the set of all elements in G which are maped to the zero element of H. For example, SL(n, R) is the kernel of the homomorphism from GL(n, R) to R 0 defined by φ(A) = det(A). \ { }
subgroup
A [subgroup] of a group G is a subset of G which is also a group. Examples:
the set of n n matrices with determinant 1 is a subgroup of the set of n n matrices with nonzero • determinant.× ×
the trivial subgroup 0 is always a subgroup of a group (G, , 0). • { } ∗ Theorem of Cauchy
The [Theorem of Cauchy] in group theory states that every finite group whose order is divisible by a prime number p contains a subgroup of order p.
[sedenions] form a zero Divisor Algebra. By a theorem of Frobenius (1877), there are three and only three associative finite division algebras: the real numbers R, the complex numbers C and the quaternions Q. Similar algebras in higher dimensions have zero divisors: sedenions are examples.
field
A [field] is a commutative ring (R, +, , 0, 1) such that (R, +, 0) and (R 0, , 1) are both commutative groups. ∗ \ ∗
theorem of Zorn
By a [theorem of Zorn] (1933), every alternative, quadratic, real non-associative algebra without zero divisors is isomorphic to the 8-dimensional octonions O.
Theorem of Hurwitz
[Theorem of Hurwitz]: the normed composition algebras with unit are: real numbers, complex numbers, quater- nions; and octonions.
Theorem of Kervaire and Milnor
[Theorem of Kervaire and Milnor] In 1958, Kervaire and Milnor proved independently of each other that the finite-dimensional real division algebras have dimensions 1, 2, 4, or 8.
Theorem of Adams
[Theorem of Adams] In 1960, Adams proved that a continuous multiplication in Rn+1 with two-sided unit and with norm product exists only for n + 1 = 1, 2, 4, or 8. Theorem of Hurwitz
[Theorem of Hurwitz]: the normed composition algebras with unit are:
real numbers • complex numbers • quaternions • octonions •
Theorems of Sylov
[Theorems of Sylov] If G is a finite group of order G = pnq, where p is a prime number, then G has a subgroup of order pn. Such groups are called Sylov groups |and| all of them are isomorphic. Furthermore, the number N of different p-Sylov groups in G satisfies N = 1 mod(p).
This file is part of the Sofia project sponsored by the Provost’s fund for teaching and learning at Harvard university. There are 39 entries in this file. Index additive, 1 algebra, 1 alternating group, 1 An algebraic number field, 1 Artinian module, 1 Artinian ring, 1
Cayley’s theorem, 2 center, 2 commutative group, 4 commutative ring, 4 commutative semigroup, 6 commutator, 2 commutator subgroup, 2
factor group, 2 field, 7 finite group, 2 function field, 4
group, 3
homomorphism, 4
isomorphism, 4
kernel, 6
normal subgroup, 3 number field, 5 octonions, 5 order, 5
p-group, 5
quaterions, 6
ring, 3
sedenions, 7 semigroup, 6 subgroup, 6
Theorem of Adams, 7 Theorem of Cauchy, 7 Theorem of Hurwitz, 7, 8 Theorem of Kervaire and Milnor, 7 theorem of Zorn, 7 Theorems of Sylov, 8
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